It is a simple rule in ancient Greek literature that mortals can never fully understand the gods and their machinations, however close some might get. The Odyssey is no exception. The mortals in it have at least two paradoxical and conflicting views of the deities they worship, which they nevertheless maintain concurrently. The gods are viewed by the men as impersonal forces to whom sacrifice must be paid, although this is no guarantee of gaining salvation. Even the suitors are seen to pay the gods their respects, even if Eumaeus denies it (at XIV.93ff.) - but he is rather biased against them. Despite their observance of the form of religious ritual, they do not understand how the gods work, and so all they do in their honour is in vain.
The second view of the gods is well displayed in the song of Demodocus about Ares and Aphrodite's affair. Here the gods are a figure of fun for mortals, acting in a far more human way than the normal visualisation of them as numinous deities would allow. This song is interesting for several reasons. Firstly, there is a hint that Homer is playing a subtle game where he plays with ideas of the truthfulness of the Bard and the reliability of his stories (we must remember that Homer was one as well). He invites us to put his own story under the same scrutiny as Demodocus' by virtue of making his tale of divine adultery hard to believe. If Demodocus' story is unlikely and without evidence, then what must we make of Homer's, which, although more plausible, cannot be corroborated either? The song also shows the paradoxical nature of Greek religion, where the serious gods can coexist with the humorous gods without difficulty, just as in the fifth century, Dionysus could be the buffoon of the Frogs or the terrifying god of the Bacchae. But the very fact that either view of the gods can be believed, when the narrative shows neither to be true (for the gods are neither omnipotent and disinterested, with the exception of Zeus, nor are they laughing stocks) is a sign that mortals cannot understand them.
Odysseus gets closest to grasping their nature, but even his idea is flawed and incomplete. He cannot even recognise a god unless that god chooses to let him do so. Hence his lies to the 'shepherd' on his arrival to Ithaca at Book XIII, for he does not realise he is talking to Athena. His understanding of the gods transcends everyone else's because he is the greatest living hero and the protagonist, and because he has been offered, and has rejected, immortality. At V.209ff., he is offered the gift of unending life by Calypso, but refuses it. He realises that he would rather live a short and difficult life and return home than eternal absence. He has deliberately rejected divinity, and this in itself brings him closer to divinity.